Scalar potential

This article is about a general description of a function used in mathematics and physics to describe conservative fields. For the scalar potential of electromagnetism, see electric potential. For all other uses, see potential.

Scalar potential, simply stated, describes the situation where the difference in the potential energies of an object in two different positions depends only on the positions, not upon the path taken by the object in traveling from one position to the other. It is a scalar field in three-space: a directionless value (scalar) that depends only on its location. A familiar example is potential energy due to gravity.

gravitational potential well of an increasing mass where F=−∇P{\displaystyle \mathbf {F} =-\nabla P}

where ∇P is the gradient of P and the second part of the equation is minus the gradient for a function of the Cartesian coordinatesx, y, z.[2] In some cases, mathematicians may use a positive sign in front of the gradient to define the potential.[3] Because of this definition of P in terms of the gradient, the direction of F at any point is the direction of the steepest decrease of P at that point, its magnitude is the rate of that decrease per unit length.

In order for F to be described in terms of a scalar potential only, any of the following equivalent statements have to be true:

−∫abF⋅dl=P(b)−P(a){\displaystyle -\int _{a}^{b}\mathbf {F} \cdot d\mathbf {l} =P(\mathbf {b} )-P(\mathbf {a} )}, where the integration is over a Jordan arc passing from location a to location b and P(b) is P evaluated at location b .

∮⁡F⋅dl=0{\displaystyle \oint \mathbf {F} \cdot d\mathbf {l} =0}, where the integral is over any simple closed path, otherwise known as a Jordan curve.

The fact that the line integral depends on the path C only through its terminal points r0{\displaystyle \mathbf {r} _{0}} and r{\displaystyle \mathbf {r} } is, in essence, the path independence property of a conservative vector field. The fundamental theorem of calculus for line integrals implies that if V is defined in this way, then F=−∇V,{\displaystyle \mathbf {F} =-\nabla V,} so that V is a scalar potential of the conservative vector field F. Scalar potential is not determined by the vector field alone: indeed, the gradient of a function is unaffected if a constant is added to it. If V is defined in terms of the line integral, the ambiguity of V reflects the freedom in the choice of the reference point r0.{\displaystyle \mathbf {r} _{0}.}

Plot of a two-dimensional slice of the gravitational potential in and around a uniform spherical body. The inflection points of the cross-section are at the surface of the body.

An example is the (nearly) uniform gravitational field near the Earth's surface. It has a potential energy

U=mgh{\displaystyle U=mgh}

where U is the gravitational potential energy and h is the height above the surface. This means that gravitational potential energy on a contour map is proportional to altitude. On a contour map, the two-dimensional negative gradient of the altitude is a two-dimensional vector field, whose vectors are always perpendicular to the contours and also perpendicular to the direction of gravity. But on the hilly region represented by the contour map, the three-dimensional negative gradient of U always points straight downwards in the direction of gravity; F. However, a ball rolling down a hill cannot move directly downwards due to the normal force of the hill's surface, which cancels out the component of gravity perpendicular to the hill's surface. The component of gravity that remains to move the ball is parallel to the surface:

FS=−mgsin⁡θ{\displaystyle F_{S}=-mg\ \sin \theta }

where θ is the angle of inclination, and the component of FS perpendicular to gravity is

However, on a contour map, the gradient is inversely proportional to Δx, which is not similar to force FP: altitude on a contour map is not exactly a two-dimensional potential field. The magnitudes of forces are different, but the directions of the forces are the same on a contour map as well as on the hilly region of the Earth's surface represented by the contour map.

In fluid mechanics, a fluid in equilibrium, but in the presence of a uniform gravitational field is permeated by a uniform buoyant force that cancels out the gravitational force: that is how the fluid maintains its equilibrium. This buoyant force is the negative gradient of pressure:

fB=−∇p.{\displaystyle \mathbf {f_{B}} =-\nabla p.\,}

Since buoyant force points upwards, in the direction opposite to gravity, then pressure in the fluid increases downwards. Pressure in a static body of water increases proportionally to the depth below the surface of the water. The surfaces of constant pressure are planes parallel to the surface, which can be characterized as the plane of zero pressure.

If the liquid has a vertical vortex (whose axis of rotation is perpendicular to the surface), then the vortex causes a depression in the pressure field. The surface of the liquid inside the vortex is pulled downwards as are any surfaces of equal pressure, which still remain parallel to the liquids surface. The effect is strongest inside the vortex and decreases rapidly with the distance from the vortex axis.

The buoyant force due to a fluid on a solid object immersed and surrounded by that fluid can be obtained by integrating the negative pressure gradient along the surface of the object:

This holds provided E is continuous and vanishes asymptotically to zero towards infinity, decaying faster than 1/r and if the divergence of E likewise vanishes towards infinity, decaying faster than 1/r2.

^The second part of this equation is only valid for Cartesian coordinates, other coordinate systems such as cylindrical or spherical coordinates will have more complicated representations, derived from the fundamental theorem of the gradient.

^See [1] for an example where the potential is defined without a negative. Other references such as Louis Leithold, The Calculus with Analytic Geometry (5 ed.), p. 1199 avoid using the term potential when solving for a function from its gradient.